Part II
Ordinary Differential Equations
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
111
II. Introduction
Differential equations arise in all fields of engineering and science. Most real
physical processes are governed by differential equations. In general, most real
physical processes involve more than one independent variable, and
corresponding differential equations are partial differential equations (PDEs).
In many cases, however, simplifying assumptions are made which reduces the
PDEs to ordinary differential equations (ODEs). This chapter is devoted to the
solution of ordinary differential equations.
II.1 General features of ordinary differential equations
An ordinary differential equation (ODE) is an equation stating a relationship
between a function sofa single independent variable and the total derivatives of
this function with respect to the independent variable.
The order of an ODE is the order of the highest-order derivative in the
differential equation. The general first-order ODE is
),( ytfdt
dy=
where f(t,y) is called the derivative function. For simplicity of notation,
differentiation usually will be denoted by the superscript "prime" notation:
dt
dyy = thus,
),( ytfy =
The general nth-order ODE for y(t) has the form
)(012
)1(
1
)( tFyayayayaya n
n
n
n =++++ −
−
The solution of an ODE is that particular function, y(t) or y(x), that
identically satisfies the ODE in the domain of interest, D(t) or D(x),
respectively, and satisfies the auxiliary conditions specified on the boundaries of
the domain of interest. In a few special cases, the solution of an ODE can be
expressed in a closed form. In the majority of problems in engineering and
science, the solution must be obtained by numerical methods. Such problems are
the subject of this chapter.
A linear ODE is one in which all of the derivatives appear in linear form and
none of the coefficients depends on the dependent variable. The coefficients
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112
may be functions of the independent variable, in which case the ODE is
variable-coefficient linear ODE. For example,
)(tFyy =+
is a linear, constant-coefficient, first-order ODE, whereas
)(tFtyy =+
is a linear, variable-coefficient, first-order ODE. If the coefficients depend on
the dependent variable, or the derivatives appear in a nonlinear form, then the
ODE is nonlinear. For example,
( ) 0
0
2=+
=+
yy
yyy
are nonlinear first-order ODEs.
A homogeneous differential equation is one in which each term involves the
dependent variable or one of its derivatives. A nonhomogenous differential
equation contain additional terms, known as nonhomogenous terms, source
terms, or forcing functions, which do not involve the dependent variable. For
example,
0=+ yy
is a linear, first-order, homogenous ODE, and
)(tFyy =+
is a linear, first-order, nonhomogenous ODE, where F(t) is the known
nonhomogenous term.
Many practical problems involve several dependent variable, each of which
is a function of the same single independent variable and one or more of the
dependent variables, and each of which is governed by an ODE. Such coupled
sets of ODEs systems of ordinary differential equations. Thus, the two coupled
ODEs
),,(
),,(
zytgz
zytfy
=
=
comprise a system of two coupled first-order ordinary differential equations.
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113
The general solution of a differential equation contains one or more constants of
integration. Thus, a family of solutions is obtained. The number of constants of
integration is equal to the order on differential equation. The particular member
of the family of solutions which is of interest is determined by auxiliary
conditions. Obviously, the number of auxiliary conditions must equal the
number of constants of integration, which is the same as the order of the
differential equation.
As illustrated in the preceding discussion, a wide variety of ODEs exists.
Each problem has its own special governing equation or equations and its own
peculiarities which must be considered individually. However, useful insights
into the general features of ODEs can be obtained by studying two special cases.
The first special case is the general nonlinear first-order ODE:
),(' ytfy =
where f(t,y) is a nonlinear function of the dependent variable y. the second
special case is the general nonlinear second-order ODE:
)(),(),( xFyyxQyyxPy =++
These two special cases are studied in the following sections.
II.2 Classification of ordinary differential equations
Physical problems are governed by many different ODEs. There are two
different types, or classes, of ODEs, depending of the type of the auxiliary
conditions specified. If all of the auxiliary conditions are specified at the same
value of the independent variable and the solution is to be marched forward
from that initial point, the differential equation is an initial-value ODE. If the
auxiliary conditions are specified at two different values of the independent
variable, the end points or boundaries of the domain of interest, the differential
equation is a boundary-value ODE.
Figure (II.1) illustrates the solution of an initial-value ODE. The initial value
of the dependent variable is specified at one value of the independent variable,
and the solution domain D(t) is open. Initial-value ODEs are solved by
marching numerical methods.
Figure (II.2) illustrates the solution of a boundary-value ODE. The boundary
values of the dependent variable are specified at two values of the independent
variable, and the solution domain D(x) is closed. Boundary-value ODEs can be
solved by both marching numerical methods and equilibrium numerical
methods.
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II.3 Classification of physical problems
Physical problems fall into one of the following three general classifications:
1. Propagation problems.
2. Equilibrium problems.
3. Eigenproblems.
Each of these three types of physical problems has its own special features, its
own particular type of ordinary differential equation, its own type of auxiliary
conditions, and its own numerical solution methods. A clear understanding of
these concepts is essential if meaningful numerical solutions are to be obtained.
Propagation problems are initial-value problems in open domains in which
the known information (initial values) are marched forward in time or space
from the initial state. The known information, that is, the initial values, are
specified at one value of the independent variable. Propagation problems are
governed by initial-value ODEs. The order of the governing ODE ma be one or
greater. The number of the initial values must be equal to the order of the
differential equation.
Propagation problems may be unsteady time (i.e., t) marching problems or
steady space (i.e., x) marching problems. The marching direction in a steady
space marching problem is sometimes called the time-like direction, and the
corresponding coordinates is called the time-like coordinate. Figure (II.3)
illustrates the open solution domains D(t) and D(x) associated with time
marching and space marching propagation problems, respectively.
Equilibrium problems are boundary-value problems in closed domains in
which the known information (boundary values) are specified at two different
values of the independent variable, the end points (boundaries) of the solution
domain. Equilibrium problems are governed by boundary-value ordinary
differential equations. The order of the governing differential equation must be
Figure (II.1) Initial-value ODE Figure (II.2) Boundary-value ODE
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
115
at least 2, and may be greater. The number of boundary values must be equal to
the order of the differential equation. Equilibrium problems are steady state
problems in closed domain. Figure (II.4) illustrates the closed solution domain
D(x) associated with equilibrium problems.
Eigenproblems are a special type of problem in which the solution exists only
for special values (i.e., eigenvalues) of a parameter of the problem. The
eigenvalues are to be determined in addition to the corresponding configuration
of the system.
Figure (II.3) Solution domain for propagation problems.
Figure (II.4) Solution domain for equilibrium problems.
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116
II.4 Initial-Value Ordinary Differential Equations
A classical example of an initial-value ODE is the general nonlinear first order
ODE:
00 )(),( ytyytfy ==
This equation applies to many problems in engineering and science. Consider
the lumped mass m illustrated in Figure (II.5). Heat transfer from the lumped
mass m to its surroundings by radiation is governed by the Stefan-Boltzmann
law of radiation:
( )44
ar TTAq −=•
where •
rq is the heat transfer rate (J/s), A is the surface area of the lumped mass
(m2), is the Stefan-Boltzmann constant (5.67 × 10-8 J/m2-K4-s), is the
emissivity of the body (dimensionless), which is the ratio of the actual radiation
to the radiation from a black body, T is the internal temperature of the lumped
mass (K), and Ta is the ambient temperature (K) (i.e., temperature of the
surroundings). The energy E stored in the lumped mass is given by
TCmE =
where m is the mass of the lumped mass (kg) and C is the specific heat of the
material (J/kg-K). An energy balance states that the rate at which the energy
stored in the lumped mass changes is equal to the rate at which heat is
transferred to the surroundings. Thus,
( ) ( )44)ar TTAq
dt
TCmd−−=−= •
The minus sign is required so that the rate of change of stored energy is negative
when T is greater than Ta. For constant m and C, the last equation can be written
as
( )mC
ATTT
dt
dTa
=−−== where44
Figure (II.5) Heat transfer by radiation from a lumped mass
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117
Consider the case where the temperature of the surroundings is constant and the
initial temperature of the lumped mass is T(0.0)=T0 . The initial-value problem
is stated as follows:
( ) 0
44 )0(),( TTTtfTTT a ==−−=
This is a nonlinear first-order initial-value ODE. The solution of this equation is
the function T(t), which describes the temperature history of the lumped mass
corresponding to the initial conditions, T(0.0)=T0 .
An example of a higher-order initial-value ODE is given by the nonlinear
second-order ODE governing the vertical flight of a rocket. The physical system
is illustrated in Figure (II.6). Applying Newton's second law of motion,
= maF , yields
yMVMMaDMgTF ===−−=
where T is the thrust developed by the rocket motor (N), M is the instantaneous
mass of the rocket (kg), g is the acceleration of gravity (m/s2), which depends on
the altitude y (m), D is the aerodynamic drag (N), a is the acceleration of the
rocket (m/s2), V is the velocity of the rocket (m/s), and y is the altitude of the
rocket (m). The initial velocity, V (0.0) = V0 , is zero, and the initial elevation,
y(0.0) = y0 , is zero. Thus, the initial conditions for the equation are
0.0)0.0(and0.0)0.0()0.0( === yyV
Figure (II.6) Vertical flight of a rocket
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118
In general the thrust T is a variable, which depends on time and altitude. The
instantaneous mass M is given by
•
−=
t
o dttmMtM0
)()(
where Mo is the initial mass of the rocket (kg), and )(tm•
is the instantaneous
mass flow rate being expelled by the rocket (kg/s). The instantaneous
aerodynamic drag D is given by
2)(2
1),,(),,( VAyyVCyVD D =
where CD is an empirical drag coefficient (dimensionless), which depends on the
rocket geometry, the rocket velocity V and the properties of the atmosphere at
altitude y (m); is the density of the atmosphere (kg/m3), which depends on the
altitude y (m); and A is the cross-sectional frontal area of the rocket (m2).
Combining the last equations yields the following second-order nonlinear initial-
value ODE:
••
−
−−
−
=t
o
D
t
o dttmM
VAyyVC
yg
dttmM
ytFy
0
2
0
)(
)(2
1),,(
)(
)(
),(
Consider a simpler model where T, •
m , and g are constant, and the aerodynamic
drag D is neglected. In that case, the last equation becomes
0.0)0.0()0.0(and0.0)0.0( ===−
−
=•
Vyyg
tmM
Fy
o
The solution of the last two equations is the function y(t), which describes the
vertical motion of the rocket as a function of time t.
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119
II.5 Boundary-Value Ordinary Differential Equations
A classical example of a boundary-value ODE is the general second-order ODE:
2211 )()()(),(),( yxyandyxyxFyyxQyyxPy ===++
This equation applies to many problems in engineering and science.
Consider the constant cross-sectional area rod illustrated in Figure (II.7). Heat
diffusion transfers energy along the rod and energy is transferred from the rod to
the surroundings by convection. An energy balance on the differential control
volume yields
)()()( xqdxxqxq c
•••
++=
which can be written as
)()()()( xqdxxqdx
dxqxq c
••••
+
+=
which yields
0)()( =+
••
xqdxxqdx
dc
Heat diffusion is governed by Fourier law of conduction, which states that
dx
dTkAxq −=
•
)(
where )(xq•
is the energy transfer rate (J/s), k is the thermal conductivity of the
solid (J/s-m-K), A is the cross-sectional area of the rod (m2), and dT/dx is the
temperature gradient (K/m). Heat transfer by convection is governed by
Newton's law of cooling:
)()( ac TThAxq −=•
where h is the empirical heat transfer coefficient (J/s-m2-K), A is the surface
area of the rod (A = P dx, m2), P is the perimeter of the rod (m), and Ta is the
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
120
ambient temperature (K) (i.e. temperature of the surroundings). From the last
three equations we can see that
( ) 0)( =−+
− aTTdxPhdx
dx
dTkA
dx
d
For constant k, A, and P, the last equation yields
( ) 02
2
=−= aTTkA
hP
dx
Td
which can be written as
aTTT 22 −=− where kA
hP=2
which is a linear second-order boundary-value ODE. The solution of this
equation is the function T(x), which describes the temperature distribution in the
rod corresponding to the boundary conditions
2211 )()( TxTandTxT ==
Figure (II.7) Steady heat conduction in a rod
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121
An example of a higher-order boundary-value ODE is given by the fourth-order
ODE governing the deflection of a laterally loaded symmetrical beam. The
physical system is illustrated in Figure (5.8). Bending takes place in the plane of
symmetry, which causes deflections in the beam. The neutral axis of the beam is
the axis along which the fibers do not undergo strain during bending. When no
load is applied (i.e. neglecting the weight of the beam itself), the neutral axis is
coincident with the x-axis. When a distributed load q(x) is applied, the beam
deflects, and the neutral axis is displaced, as illustrated by the dashed line in
Figure (II.8). The shape of the neutral axis is called the deflection curve.
As shown in many strength of materials books (e.g. Timoshenko, 1955), the
differential equation of the deflection curve is
)()(2
2
xMdx
ydxIE −=
where E is the modulus of elasticity of the beam material, I(x) is the moment of
inertia of the beam cross-section, which can vary along the length of the beam,
and M(x) is the bending moment due to transverse forces on the beam, which
can vary along the length of the beam. The moment M(x) is related to the
shearing forces V(x) acting on each cross-section of the beam as follows:
)()(
xVdx
xdM=
The shearing forces V(x) is relate to the distributed load q(x) as follows:
)()(
xqdx
xdV−=
Combining the three equations yields the differential equation for the beam
deflection curve:
)()(4
4
xqdx
ydxIE =
This equation requires four boundary conditions. For a horizontal beam of
length L,
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122
For a supported beam at both ends (any kind of rigid support):
0.0)()0.0( == Lyy
For a beam fixed (i.e. clamped) at both ends:
0.0)()0.0( == Lyy
For a beam pinned (i.e. hinged) a both ends:
0.0)()0.0( == Lyy
For a beam cantilevered (i.e. free) a either ends:
0.0)(or0.0)0.0( == Lyy
Any two combinations of these four boundary conditions can be specified at
each end.
The last equation is a linear example of the general nonlinear fourth-order
boundary-value ODE:
),,,,( yyyyxfy =
which requires four boundary conditions at the boundaries of the closed physical
domain.
Figure (II.8) Deflection of a beam
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123
Chapter 5
One-Dimensional
Initial-Value Ordinary
Differential Equations
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124
5. One-Dimensional Initial-Value Ordinary Differential Equations
The initial-value ODEs govern propagation problems, which are initial-value
problems in open domains. Consequently, initial-value ODEs are solved
numerically by marching methods. This section is devoted to presenting the
basic properties of finite difference methods for solving initial-value (i.e.
propagation) problems and to developing several specific finite difference
methods.
The objective of a finite difference method for solving an ordinary
differential equation (ODE) is to transform a calculus problem into an algebra
problem by:
1. Discretizing the continuous physical domain into a discrete finite
difference grid.
2. Approximating the exact derivatives in the ODE by algebraic finite
difference approximations (FDAs).
3. Substituting the FDAs into the ODE to obtain an algebraic finite
difference equation (FDE).
4. Solving the resulting algebraic FDE.
5.1 General Features of Initial-Value ODEs 5.1.1 The general linear first-order ODE
The general linear first-order ODE is given by
00 )()( ytytFyy ==+
where α is a real constant, which can be positive or negative.
The exact solution of this equation is the sum of the complementary
solution )(tyc and the particular solution )(tyP :
)()()( tytyty pc +=
The complementary solution )(tyc is the solution of the homogeneous ODE:
0=+ cc yy
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125
The complementary solution, which depends only on the homogeneous ODE,
describes the inherent properties of the ODE and the response of the ODE in the
absence of external stimuli. The particular solution yp(t) is the function which
satisfies the nonhomogeneous term F(t):
)(tFyy pp =+ …………...…………………………………………… (1)
The particular solution describes the response of the ODE to external stimuli
specified by the nonhomogeneous term F(t).
The complementary solution yc(t) is given by
t
c eAty −=)(
which can be shown to satisfy the homogeneous equation by direct substitution.
The coefficient A can be determined by the initial condition, y(t0) = y0 , after the
complete solution y(t) to the ODE gas been obtained. The particular solution
yp(t) is given by
...)()()()( 21 +++= tFBtFBtFBty op
where the terms F'(t), F''(t), etc. are the derivatives of the nonhomogeneous
term. These terms BiF(i)(t) , continue until F(i)(t) repeats its functional form or
becomes zero. The coefficients B1 , B2 , etc. can be determined by substituting
the last equation into equation (1), grouping similar fractional forms, and
requiring that the coefficient of each fractional form be zero so that equation (1)
is satisfied for all values of independent variable t.
The total solution is then obtained as
)()( tyeAty p
t += −
The constants of integration A can be determined by requiring the last equation
to satisfy the initial condition, y(t0) = y0 .
The homogeneous ODE, 0' =+ yy , has two completely different types of
solutions, depending on whether α is positive or negative. Consider the pair of
ODEs:
0'
0'
=−
=+
yy
yy
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
126
where α is a positive real constant. The solutions of these two equations are
t
t
eAty
eAty
=
= −
)(
)(
Each of these two equations specify a family of solutions, as illustrated in
Figure (5.1). A particular member of either family is chosen by specifying the
initial conditions, y(0) = y0 , as illustrated by the dashed curves in Figure (5.1).
For the ODE 0' =+ yy , the solution, teAty −=)( , decays exponentially
with time. This is a stable ODE. For the ODE 0' =− yy , the solution, teAty =)( , grows exponentially with time without bound. This is an unstable
ODE. Any numerical method for solving these ODEs must behave in a similar
manner.
Figure (5.1) Exact solutions of first-order homogeneous
ODEs. (a) 0' =+ yy . (b) 0' =− yy
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
127
5.1.2 The general nonlinear first-order ODE
The general nonlinear first-order ODE is given by
00 )(),( ytyytfy == ……..……………………………………….. (2)
where the derivative function f(t,y) is a nonlinear function of y. the general
features of the solution of a nonlinear first-order ODE in a small neighborhood
of a point are similar to the general features of the solution of the linear first-
order ODE at that point.
The nonlinear ODE can be linearized by expressing the derivative function f(t,y)
in a Taylor series at an initial point (t0 , y0) and dropping all terms higher than
first order. Thus,
( ) ...),(
...)()(),(
0000000
00000
+++−−=
+−+−+=
yftfyftffytf
yyfttffytf
ytyt
yt
which can be written as
termserhigher ordtFyytf ++−= )(),(
where α and F(t) are defined as follows:
( ) tfyftfftF
f
tyt
y
000000
0
)( +−−=
−=
Truncating the higher-order terms and substituting the last three equations into
the nonlinear ODE, Eq. (2), gives
00 )()(' ytytFyy ==+
Most of the general features of the numerical solution of a nonlinear ODE can
be determined by analyzing the linearized form of the nonlinear ODE.
Consequently, the linearized equation will be used extensively as a model ODE
to investigate the general behavior of numerical methods for solving initial-
value ODEs, both linear and nonlinear.
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
128
5.1.3 Higher-Order ODEs
Higher-order ODEs generally can be replaced by a system of first-order ODEs.
Each higher-order ODE in a system of higher-order ODEs can be replaced by a
system of first-order ODEs, thus yielding coupled systems of first-order ODEs.
5.1.4 Systems of first-order ODEs
Systems of coupled first-order ODEs can be solved by solving all the coupled
first-order differential equations simultaneously using methods developed for
solving single first-order ODEs.
5.2 The Taylor Series Method
In theory, the infinite Taylor series can be used to evaluate a function, given its
derivative function and its value at some point. Consider the nonlinear first-
order ODE:
00 )(),( ytyytfy ==
The Taylor series for y(t) at t = t0 is
......))((1
......
))((6
1))((
2
1))(()()(
00
)(
3
00
2
00000
+−++
−+−+−+=
nn tttyn
tttytttytttytyty
which can be written in simpler appearing form:
......6
1
2
1)( 3
0
2
000 ++++= tytytyyty
This equation can be employed to evaluate y(t) if y0 and the values of the
derivatives at t0 can be determined. The value of y0 is the initial condition. The
first derivative 0
'y can be determined by evaluating the derivative function f(t,y)
at t0 : ),(' 000ytfy = . The higher-order derivatives can be determined by
successively differentiating the lower-order derivatives, starting with y'. Thus,
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
129
( ) ( )
( ) 22)(2
)()y(y
manner,similaraIn
y
.thatRecall
)(
)),(()(
)()(
yyyyyyyyyy
dt
dyyyy
yyyy
tdt
yd
yyy
ydt
dy
dt
dyyy
dt
dy
y
y
t
y
dt
yd
dt
dy
y
y
t
ydtdy
y
ydt
t
yytydyd
dt
ydyy
yyyyttytt
ytyt
yt
yt
++++=
+
++
=
==
+=
=
+=
+
=
+
=
+
==
==
Higher-order derivatives become progressively more complicated. It is not
practical to evaluate a large number of the higher-order derivatives.
Consequently, the Taylor series must be truncated. The remainder term in a
finite Taylor series is:
Remainder ( )
( ) 1)1(
!1
1 ++ +
= nn tyn
where tt 0 . Truncating the remainder term yields a finite truncated Taylor
series. Error estimation is difficult since is unknown.
Example: The Taylor series method
Let's solve the radiation problem by the Taylor series method.
0.2500.2500)0.0()(),(' 44 ==−−== aa TTTTTtfT
where α = 4.0×10-12(K3-s)-1. The Taylor series for T(t) is given by
......24
1'''
6
1''
2
1')( 4
0
)4(3
0
2
000 +++++= tTtTtTtTTtT
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
130
where tttt =−= 0 . From initial conditions, T0 = 2500.0, and
( )( ) 234375.1560.2500.2500100.4)(' 4412
0
44
0−=−−=−−= −
aTTT
Solving for the higher-order derivatives yields:
( )
( ) ( )
( )
( ) ( )
( ) ( )( )
( )
( ) ( )
( )( )
432
8
459312)4(
84593)4(
8246103
426212
0
4262
4372
312
0
3
444965.0847900.2529297.19234375.156-2500.0T(t)
:yieldsofseriesTaylorinvaluesabovethemgSubstituti
679169.10234375.156)0.2500.25006
0.2500.2500600.250070(100.44
6607040.0
31074
087402.17
234375.1560.2500.250030.25007100.44
3740.0
4
058594.39234375.1560.2500100.44
40.0
tttt
T(t)
T
TTTTTTTT
T
t
TTT
TTTTTT
T
TTTTTT
T
t
TTT
TTTT
T
TTTT
T
t
TTT
aa
aa
a
a
+−+=
=−
+−−=
+−−=
+
=
=
+−−=
−=
−−=
−+=
+
=
=
−=
=−−=
−=
+
=
=
−
−
−
The exact solution and the solution obtained from the above equation are
tabulated in the following table, where )2()1( )(,)( tTtT , etc. denote the Taylor
series through the first, second, etc. derivative terms. These results are also
illustrated in Figure (5.2).
From Figure (5.2), it is obvious that the accuracy of the solution improves as the
number of terms in the Taylor series increases. However, even with four terms,
the solution is not very accurate for t > 2.0 s. The Taylor series method is not an
efficient method for solving initial-value ODEs.
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
131
Even though the Taylor series method is not an efficient method for solving
initial-value ODEs, it is the basis of many excellent numerical methods. As
illustrated in Figure (5.10), the solution by the Taylor series method is quite
accurate for small value of t. Therein lies the basis for more accurate methods of
solving ODEs. Simply put, use the Taylor series method for a small step in
neighborhood of the initial point. Then reevaluate the coefficients (i.e. the
derivatives) at the new point. Successive reevaluation of the coefficients as the
solution progresses yields a much more accurate solution. This concept is the
underlying basis of most numerical methods for solving initial-value ODEs.
Figure (5.2) Solution by the Taylor series method.
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
132
5.3 The Finite Difference Method 5.3.1 Finite difference grids
The solution domain D(t) [or D(x)] and a discrete finite difference grid are
illustrated in Figure (5.3). The solution domain is discretized by a one-
dimensional set of discrete grid points, which yields the finite difference grid.
The finite difference solution of the ODE is obtained at these grid points. For
the present, let these grid points be equally spaced having uniform spacing t (or x ). The resulting finite difference grid is illustrated in Figure (5.3). The
subscript n is used to denote the physical grid points, that is, tn (or xn). Thus,
grid point n corresponds to location tn (or xn) in the solution domain D(t) [or
D(x)]. The total number of grid points is denoted by nmax.
The dependant variable at grid point n is denoted by the same subscript
notation that is used to denote the grid points themselves. Thus, the function y(t)
at grid point n is denoted by
nn yty =)(
In a similar manner, derivatives are denoted by
nnn ytytdt
dy== )()(
Figure (5.3) Solution domain, D(t) [or D(x)], and discrete finite difference grid.
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
133
5.3.2 Finite difference approximations
Now that the finite difference grid has been specified, finite difference
approximations (FDAs) of the exact derivatives in the ODE must be developed.
This is accomplished using the Taylor series approach developed in Chapter 4.
In the development of finite difference approximations of differential
equations, a distinction must be made between the exact solution of the
differential equation and the solution of the finite difference equation which is
an approximation of the exact differential equation. For the remainder of this
chapter, the exact solution of the ODE is denoted by an overbar on the symbol
for the dependent variable [i.e. )(ty ], and the approximate solution is denoted
by the symbol for the dependent variable without an overbar [i.e. )(ty ]. Thus,
solutioneapproximat)(
solutionexact)(
=
=
ty
ty
Exact derivatives, such as y , can be approximated at a grid point in terms of
the values of y at that grid point and adjacent grid points in several ways.
Consider the derivative y . Writing the Taylor series for 1+ny using grid point n
as the base point gives
.....6
1
2
1 32
1 ++++=+ tytytyyynnnnn
This equation can be expressed as the Taylor polynomial with remainder:
1)(2
1!
1.....
2
1 +
+ +++++= mm
n
m
nnnn Rtym
tytyyy
where the remainder Rm+1 is given by
1)1(1 )(!)1(
1 +++ +
= mmm tym
R
where ttt + . The remainder term is simply the next term in Taylor series
evaluated at t = τ. If the infinite Taylor series is truncated after the mth derivative
term to obtain an approximation of 1+ny , the remainder term Rm+1 is the error
associated with the truncated Taylor series. In most cases, our main concern is
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
134
the order of the error, which is the rate at which the error goes to zero
as 0→t .
Solving the last equation for n
y yields
......6
1
2
1 21 −−−
−= + tyty
t
yyy
nn
nn
n
If this equation is terminated after the first term on the right-hand side, it
becomes
tyt
yyy nn
n−
−= + )(
2
11
A finite difference approximation of n
y , which will be denoted by n
y , can be
obtained from the last equation by truncating the remainder term. Thus,
where )(0 t term is shown to remind us of the order of the remainder term
which was truncated, which is the order of the approximation of n
y . The
remainder term which has been truncated to obtain the last equation is called the
truncation error of the finite difference approximation of n
y . This equation is
a first-order forward-difference approximation of yat grid point n.
A first-order backward-difference approximation of y at grid point n+1
can be obtained by writing the Taylor series for ny using grid point n+1 as the
base point and solving for 1+
n
y . Thus,
( ) ( )
tyt
yyy
tytyyy
nn
n
nnnn
+
−=
+−+−+=
+
+
+++
)(2
1
.....2
1
1
1
2
111
Truncating the remainder term yields
)(01
1t
t
yyy nn
n
−= +
+
)(01 tt
yyy nn
n
−= +
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
135
A second-order centered-difference approximation of y at grid point 2
1+n can
be obtained by writing the Taylor series for nn yandy 1+ using grid point 2
1+n
as the base point, subtracting the two Taylor series, and solving for 2
1+
n
y .
Thus,
......26
1
22
1
2
......26
1
22
1
2
3
2
1
2
2
1
2
1
2
1
3
2
1
2
2
1
2
1
2
11
+
−+
−+
−+=
+
+
+
+=
++++
+++++
ty
ty
tyyy
ty
ty
tyyy
nnnnn
nnnnn
Subtracting the second equation from the first one, and solving for 2
1+
n
y yields
( ) 21
2
124
1ty
t
yyy nn
n−
−= +
+
Truncating the remainder term yields
Note that the three (forward-backward-centered) equations of y are identical
algebraic expressions. They all yield the same numerical value. The differences
in the three finite difference approximations are the value of the truncation
errors.
All the above equations can be applied to steady space marching problems
simply by changing t to x in all the equations.
Occasionally a finite difference of an exact derivative is presented without
its development. In such cases, the truncation error and order can be determined
by a consistency analysis using Taylor series. For example, consider the
following finite difference approximation (FDA):
t
yyFDA nn
−= +1
( )21
2
1 0 tt
yyy nn
n
−= +
+
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
136
The Taylor series for the approximate solution y(t) with base point n is
......2
1 2
1 +++=+ tytyyy nnnn
Substituting the Taylor series for yn+1 into the FDA, yields
......2
1......
2
1 2
++=
−+++= tyy
t
ytytyyFDA nn
nnnn
As 0→t , FDA → ny , which shows that FDA is an approximation of the
exact derivative y at grid point n. The order of FDA is 0(∆t). The exact form of
the truncation error relative to grid point n is determined. Choosing other base
points for the Taylor series yields the truncation errors relative to those base
points.
A finite difference approximation (FDA) of an exact derivative is
consistent with the exact derivative if the FDA approaches the exact derivative
as 0→t , as illustrated in the last equation. Consistency is an important
property of finite difference approximation of derivatives.
5.3.3 Finite difference equations
Finite difference solutions of differential equations are obtained by discretizing
the continuous solution domain and replacing the exact derivatives in the
differential equation by finite difference approximations to obtain a finite
approximation of the differential equation. Such approximations are called finite
difference equations (FDEs).
Consider the general nonlinear initial-value ODE:
0)0(),( yyytfy ==
Choose a finite difference approximation (FDA), y , for y . For example:
t
yyy
t
yyy nn
nnn
n
−=
−= +
++ 1
11 or
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
137
Substitute the FDA for y into the exact ODE, ),( ytfy = , and solve for yn+1:
1111
1
1
),(
),(
++++
+
+
==
−=
==
−=
nnnnn
n
nnnnn
n
fytft
yyy
fytft
yyy
Solving the first equation for yn+1 yields
nnnnnn ftyytftyy +=+=+ ),(1 ………………………………….. (1)
Solving the second equation for yn+1 yields
1111 ),( ++++ +=+= nnnnnn ftyytftyy ……………………………… (2)
Equation (1) is an explicit finite difference equation, since fn does not depend on
yn+1 , and it can be solved explicitly for yn+1. Equation (2) is an implicit finite
difference equation, since fn+1 depends on yn+1 . If the ODE is linear, them fn+1 is
linear in yn+1 , and it can be solved directly for yn+1 . If the ODE is nonlinear,
then fn+1 us nonlinear in yn+1 , and additional effort is required to solve it for yn+1.
5.3.4 Smoothness
Smoothness refers to the continuity of a function and its derivatives. The finite
difference method of solving a differential equation employs Taylor series to
develop finite difference approximations (FDAs) of the exact derivatives in the
differential equation. If a problem has discontinuous derivatives of some order at
some point in the solution domain, then FDAs based on the Taylor series may
misbehave at that point.
For example, consider the vertical flight of a rocket illustrated in Figure
(II.6). When the rocket engine is turned off, the thrust drops to zero instantly.
This causes a discontinuity in the acceleration of the rocket, which causes
a discontinuity in the second derivative of the altitude y(t). The solution is not
smooth in the neighborhood of the discontinuity in the second derivative of y(t).
At a discontinuity, single point methods or extrapolation methods should be
employed, since the step size in the neighborhood of the discontinuity can be
chosen so that the discontinuity occurs at a grid point. Multipoint methods
should not be employed in the neighborhood of a discontinuity in the function or
its derivatives.
Problems which do not have any discontinuities in the function or its
derivatives are called smoothly varying problems. Problems which have
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
138
discontinuities in the function or its derivatives are called nonsmoothly varying
problems.
5.3.5 Errors
Five types of errors can occur in the numerical solution of differential equations:
1. Errors in the initial data (assumed nonexistent).
2. Algebraic errors (assumed nonexistent).
3. Truncation errors.
4. Roundoff errors.
5. Inherited errors.
A differential equation has an infinite number of solutions, depending on the
initial conditions. Thus, a family of solutions exists for the linear first-order
homogenous ODEs as illustrated in Figure (5.4). Figure (5.4-a) illustrates a
family of converging solutions for a stable ODE, and Figure (5.4-b) illustrates a
family of diverging solutions for an unstable ODE. An error in the initial
conditions or an algebraic error simply moves the solution to a different member
of the solution family. Such errors are assumed to be nonexistent.
Any error in the numerical solution essentially moves the numerical
solution to a different member of the solution family. Consider the converging
family of solutions illustrated in Figure (5.4-a). Since the members of the
solution family converge as t increases, errors in the numerical solution of any
type tend to diminish as t increases. By contrast, for the diverging family of
solutions illustrated in Figure (5.4-b), errors in the numerical solution of any
type tend to grow as t increases.
Truncation error is the error incurred in a single step caused by truncating
the Taylor series approximations for the exact derivatives. Truncation error
depends on the step size- 0(∆tn). Truncation error decreases as the step ∆t
decreases. Truncation errors propagate from step to step and accumulate as the
number of steps increases.
Round-off error is the error caused by the finite word length employed in
the calculation. Round-off error is more significant when small differences
between large numbers are calculated. Consequently, round-off error increases
as the step size ∆t decreases, both because the changes in the solution are
smaller and more steps are required. Most computers have 32 bit or 64 bit word
length, corresponding to approximately 7 or 13 significant decimal digits,
respectively. Some computers have extended precision capability, which
increases the number of bits to 128. Care must be exercised to ensure that
enough significant digits are maintained in numerical calculations so that round-
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
139
off is not significant. Round-off errors propagate form step to step and tend to
accumulate as the number of calculations (i.e. steps) increases.
Inherited errors is the sum of all accumulated errors from all previous
steps. The presence of inherited error means that the initial condition for the next
step is incorrect. Essentially, each step places the numerical solution on a
different member of the solution family. Assuming that algebraic errors are
nonexistent and that round-off errors are negligible, inherited error is the sum of
all previous truncation errors. On the first step, the total error at the first solution
point is the local truncation error. The initial point for the second step is on a
different member of the solution family. Another truncation error is made on the
second step. This truncation error is relative to the exact solution passing
through the first solution point. The total error at the second solution point id
due both to the truncation error at the first solution point, which is now called
inherited error, and the local truncation error of the second step. This dual error
source, inherited error and local truncation error, affects the solution at each
step. For a converging solution family, inherited error remains bounded as the
solution progresses. For a diverging solution family, inherited error tends to
grow as the solution progresses. One practical consequence of these effects is
that smaller step sizes may be required when solving unstable ODEs which
govern converging solution families.
Figure (5.4) Numerical solutions of the first-order linear homogeneous ODEs.
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
140
5.4 The First-Order Euler Methods
The explicit Euler method and the implicit Euler method are two first-order
finite difference methods for solving initial-value ODEs. Although these
methods are too inaccurate to be of much practical value, they are useful to
illustrate many concepts relevant to the finite difference solution of initial-value
ODEs.
5.4.1 The Explicit Euler Method
Consider the general nonlinear first-order ODE:
00 )(),( ytyytfy ==
Choose point n as the base point and develop a finite difference approximation
of this equation at that point. The finite difference grid is illustrated in
Figure (5.5), where the cross (i.e. ×) denotes the base point for the finite
difference approximation of the equation. The first-order forward-difference
finite difference approximation of y is given previously by
tyt
yyy nn
n−
−= + )(
2
11
Substituting this equation in the general nonlinear first-order ODE and
evaluating ),( ytf at point n yields
nnnnnn fytfty
t
yy==−
−+ ),()(2
11
Solving for 1+ny gives
Figure (5.5) Finite difference grid for the explicit Euler method
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
141
)(0)(2
1 22
1 tftytyftyy nnnnnn ++=++=+
Truncating the remainder term, which is )(0 2t , and solving for yn+1 yields the
explicit Euler finite difference equation (FDE):
where the )(0 2t term is included as a remainder of the order of the local
truncation error. Several features of this equation is summarized below:
1. The FDE is explicit, since fn does not depend on yn+1.
2. The FDE requires only one known point. Hence, it is a single
point method.
3. The FED requires only one derivative function evaluation [i.e. f(t,y)]
per step.
4. The error in calculating yn+1 for a single step, the local truncation error,
is )(0 2t .
5. The global (i.e. total) error accumulated after N steps is 0(∆t). This result
is derived in the following paragraph.
The explicit Euler finite difference equation is applied repetitively to march
from the initial point t0 to the final point, tN , as illustrated in Figure (5.6). The
solution at point N is
( ) −
=
+
−
=
+ +=−+=1
0
10
1
0
10
N
n
n
N
n
nnN yyyyyy
The total truncation error is given by
21
0
2
0 )(2
1)(
2
1Error tyNtyy
N
n
n =
+=
−
=
where Ntt 0 . The number of steps N is related to the step size ∆t as
follows:
t
ttN N
−= 0
)(0 2
1 tftyy nnn +=+
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
142
Substituting the last equation in the error equation yields
( ) )(0)(2
1Error 0 ttyttN =−=
Consequently, the global (i.e. total) error of the explicit Euler FDE is 0(∆t),
which is the same as the order of the finite difference approximation of the exact
derivative y , which is 0(∆t), as shown previously.
The result developed in the preceding paragraph applies to all finite
difference approximations of first-order ordinary differential equations. The
order of the global error is always equal to the order of the finite difference
approximation of the exact derivative y .
The algorithm base on the repetitive application of the explicit Euler FEE
to solve initial-value ODEs is called the explicit Euler method.
Example: The explicit Euler method
Let's solve the radiation problem presented earlier using the explicit Euler finite
difference equation. The derivative function is ( )44),( aTTTtf −−= . Thus,
( )44
1 annn TTtTT −−=+
Let st 0.2= . For the first time step,
Figure (5.6) Repetitive application of the explicit Euler method.
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
143
( )( )( ) 531250.2187234375.1560.20.2500
234375.1560.2500.2500100.4
1
4412
0
=−+=
−=−−= −
T
f
The results and the result of the subsequent time steps for t from 4.0s to 10.0s
are summarized in the following table. The results for ∆t =1.0s are also
presented.
Several important features of the explicit Euler method are illustrated in that
table. First, the solutions for both step sizes are following the general trend of
the exact solution correctly. The solution for the smaller step size is more
accurate than the solution for the larger step size. In fact, the order of the method
can be estimated by comparing the errors at t = 10.0s. Thus,
( ) ( ) ( )( )
( ) ( ) ( )( )0.12
10.1
0.22
10.2
0
0
TtttE
TtttE
N
N
−==
−==
Assuming that the value of ( )T are approximately equal, the ratio of the
theoretical error is
( )( )
15.2629260.28
515406.61
0.1
0.2Ratio =
−
−=
=
==
tE
tE
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
144
The ratio shows that the method is first order. The value of 2.15 is not exactly
equal to the theoretical value 2.0 due to the finite step size. The theoretical value
of 2.0 is achieved only in the limit as 0→t .
Another feature illustrated in the table is that the errors are relatively large.
This is due to the large first-order, 0(∆t), truncation error. The errors are all
negative, indicating that the numerical solution leads to the exact solution. This
occurs because the derivative function f(t,T) decreases as t increases as
illustrated in the table. The derivative function in the FDE is evaluated at point
n, the beginning of the interval of integration, where it has its largest value for
the interval. Consequently, the numerical solution leads the exact solution.
The final feature of the explicit Euler method which is illustrated in the
table is that the numerical solution approaches the exact solution as the step size
decreases. This property of a finite difference method is called convergence.
Convergence is necessary for a finite difference method to be of any use in
solving a differential equation.
________________________________________________________________
When the base point for the finite difference approximation of an ODE is point
n, the unknown value yn+1 appears in the finite difference approximation of y ,
but not in the derivative function ),( ytf . Such FDEs are called explicit FDEs.
The explicit Euler method is the simplest example of an explicit FDE.
When the base point for the finite difference approximation of an ODE is
point n+1, the unknown value yn+1 appears in the finite difference approximation
of y and also in the derivative function ),( ytf . Such FDEs are called implicit
FDEs.
5.4.2 The Implicit Euler Method
Consider the general nonlinear first-order ODE:
00 )(),( ytyytfy ==
Figure (5.7) Finite difference grid for the implicit Euler method.
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
145
Choose point n+1 as the base point and develop a finite difference
approximation of the above general nonlinear first-order ODE at that point. The
finite difference grid is illustrated in Figure (5.7). The first-order backward
difference finite difference approximation of y is given previously by:
tyt
yyy n
nn
n+
−=
++
+)(
2
11
1
1
Substituting this equation into the general nonlinear first-order ODE, and
evaluating ),( ytf at point n+1 yields
( ) 11111 ,)(
2
1++++
+ ==+
−nnnn
nn fytftyt
yy
Solving for 1+ny gives
( )2
1
2
111 0)(2
1tftytyftyy nnnnnn ++=−+= ++++
Truncating the ( )20 t remainder term yields the implicit Euler FDE:
Several features of this equation are summarized below:
1. The FDE is implicit, since fn+1 depends on yn+1. If f(t,y) is linear in y, then
fn+1 is linear in yn+1, and this equation is a linear FDE which can be solved
directly for yn+1. If f(t,y) is nonlinear in y, then the equation is a nonlinear
FDE, and additional effort is required to solve for yn+1.
2. The FDE is a single-point FDE.
3. The FDE requires only one derivative function evaluation per step if f(t,y)
is linear in y. If f(t,y) is nonlinear in y, the equation is nonlinear in yn+1,
and several evaluations of the derivative function may be required to
solve the nonlinear FDE.
4. The single-step truncation error is ( )20 t , and the global error is ( )t0 .
( )2
11 0 tftyy nnn += ++
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
146
The algorithm based on repetitive application of the implicit Euler FDE to solve
initial-value ODEs us called the implicit Euler method.
The derivative function ),( ytf may be linear or nonlinear in y . When
),( ytf is linear in y , the corresponding FDE is linear in yn+1, for both
explicit FDEs and implicit FDEs. When ),( ytf is nonlinear in y , explicit
FDEs are still linear in yn+1. However, implicit FDEs are nonlinear in yn+1, and
special procedures are required to solve for yn+1. One of the procedures
(Newton's method) is discussed in the next example.
Example: The implicit Euler method.
Let's solve the radiation problem presented earlier using the implicit Euler finite
difference equation. The derivative function is ( )44),( aTTTtf −−= . Thus,
( )44
11
11
annn
nnn
TTtTT
ftTT
−−=
+=
++
++
This equation is a nonlinear fourth-order polynomial FDE. Procedure for
solving nonlinear implicit FDEs is presented in the following using Newton's
method.
Rearranging the last equation into the form of )( 11 ++ = nn yGy or,
( ) 0)( 111 =−= +++ nnn yGyyF
Expanding )( 1+nyF in a Taylor series about the value yn+1 and evaluating at 1~
+ny
yields
( ) ( ) ( )( ) 0.......~~11111 =+−+= +++++ nnnnn yyyFyFyF
where 1~
+ny is the solution of )( 11 ++ = nn yGy . Truncating the last equation after
the first-order term and solving for yn+1 yields
( )( ))(
1
)(
1)(
1
)1(
1 k
n
k
nk
n
k
nyF
yFyy
+
++
+
+
−=
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
147
The last equation must be solved iteratively. Newton's method works well for
nonlinear implicit FDEs. A good initial guess may be required.
Returning back to our example where:
( )44
11
11
annn
nnn
TTtTT
ftTT
−−=
+=
++
++
Rearranging the equation to be solved using Newton's method yields
( ) ( ) 044
111 =−+−= +++ annnn TTtTTTF
The derivative of ( )1+nTF is
( ) 3
11 41 ++ +=nn TtTF
Then
( )( ))(
1
)(
1)(
1
)1(
1 k
n
k
nk
n
k
nTF
TFTT
+
++
+
+
−=
Let st 0.2= . For the first time step,
( ) ( )( )( )( ) ( )( ) 3
1
12
1
4412
11
100.40.241
0.2500.2500100.40.20.2500
TTF
TTF
−
−
+=
−+−=
Let KT 0.2500)0(
1 = . Then
( ) ( )( )( )
( ) ( )( )
687500.2291500000.1
468250.3120.2500
500000.10.2500100.40.241
468250.312
0.2500.2500100.40.20.25000.2500
)1(
1
312)0(
1
4412)0(
1
=−=
=+=
=
−+−=
−
−
T
TF
TF
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
148
Repeating the procedure three more times yields the converged result
785819.2282)4(
1 =T . These results are presented in the following table, along
with the final results for the subsequent time steps from t = 4.0s to 10.0s. The
results for ∆t =1.0s are also presented.
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
149
The results presented in the last table behave generally the same as the results
presented in the explicit Euler method. An error analysis at t = 10.0s gives
( )( )
90,1468684.25
455617.48
0.1
0.2Ratio ==
=
==
tE
tE
which shows that the method is first order. The errors are all positive, indicating
that the numerical solution lags the exact solution. This result is in direct
contrast to the error behavior of the explicit Euler method, where a leading error
was observed. In the present case, the derivative function of the FDE is
evaluated at point n+1, the end of the interval of integration, where it has its
smallest value. Consequently, the numerical solution lags the exact solution.
5.4.3 Comparisons of the Explicit and Implicit Euler Methods
The explicit Euler method and the implicit Euler method are both first-order
[i.e. 0(∆t)] methods. As illustrated in the last two examples, the errors in these
two methods are comparable (although of opposite sign) for the same step size.
For nonlinear ODEs, the explicit Euler method is straightforward, but the
implicit Euler method yields a nonlinear FDE, which is more difficult to solve.
So what is the advantage, if any, of the implicit Euler method?
The implicit Euler method is unconditionally stable, whereas the explicit
Euler method is conditionally stable. This difference can be illustrated by
solving the linear first order homogeneous ODE
1)0(0 ==+ yyy
For which yytf −=),( , by both methods. The exact solution is
tety −=)(
Solving the ODE by the explicit Euler method yields the following FDE:
( )nnnnn ytyftyy −+=+=+1
( ) nn yty −=+ 11
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
150
Solutions of this equation for several values of ∆t are presented in Figure (5.8).
The numerical solution behave in a physically correct manner (i.e. decrease
monolithically) for → tt as0.1 , and approaches the exact asymptotic
solution, 0)( =y . For 0.1=t , the numerical solution reaches the exact
asymptotic solution, 0)( =y , in one step.
For 0.20.1 t , the numerical solution overshoots and oscillates about
the exact asymptotic solution, 0)( =y , in a damped manner and approaches
the exact asymptotic solution as →t . For 0.2=t , the numerical solution
oscillates about the exact asymptotic solution in a stable manner but never
approaches the exact asymptotic solution. Thus, solutions are stable
for 0.2t .
For 0.2t , the numerical solution oscillates about the exact asymptotic
solution in an unstable manner that grows exponentially without bound. This is
numerical instability. Consequently, the explicit Euler method is conditionally
stable for this ODE, that is, it is stable only for 0.2t .
The oscillatory behavior for 0.20.1 t is called overshoot and must be
avoided. Overshoot is not instability. However, it does not model physical
reality, thus it is unacceptable. The step size ∆t generally must be 50 percent or
less of the stable step size to avoid overshoot.
Figure (5.8) Behavior of the explicit Euler method
Engineering Numerical Methods Chapter 5: Numerical solutions of initial value ODEs
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Solving the ODE by the implicit Euler method gives the following FDE:
( )111 +++ −+=+= nnnnn ytyftyy
This equation is linear in yn+1, it can be solve directly for yn+1 to yield
Which can be solved for several values of ∆t as presented in Figure (5.9). The
numerical solution behave in a physically correct manner (i.e decrease
monotonically) for all values of ∆t. This is unconditional stability, which is the
main advantage of implicit methods. The error increase as ∆t increases, but this
is an accuracy problem, not a stability problem.
t
yy n
n+
=+1
1
Figure (5.9) Behavior of the implicit Euler method